EP4655921A1

EP4655921A1 - Method of receiving a transmitted signal over a time-varying frequency-selective channel and receiver thereof

Info

Publication number: EP4655921A1
Application number: EP24747526.2A
Authority: EP
Inventors: Xiaobei LIU; Yong Liang Guan
Original assignee: Nanyang Technological University
Current assignee: Nanyang Technological University
Priority date: 2023-01-25
Filing date: 2024-01-24
Publication date: 2025-12-03
Also published as: WO2024158340A1; EP4655921A4

Abstract

A method of receiving a transmitted signal over a time-varying frequency-selective channel is provided. The method includes: obtaining a received symbol signal in frequency domain based on the transmitted signal; performing a pilot-aided channel estimation with respect to the time- varying frequency-selective channel using a first trained neural network model based on the received symbol signal to obtain a plurality of first estimated channel coefficients; performing a first equalization based on the received symbol signal and the plurality of first estimated channel coefficients to obtain a plurality of first detected source symbols; performing a data- aided channel estimation with respect to the time-varying frequency-selective channel using a second trained neural network model based on the received symbol signal and the plurality of first detected source symbols to obtain a plurality of second estimated channel coefficients; and performing a second equalization based on the plurality of second estimated channel coefficients to obtain a plurality of second detected source symbols. There is also provided a corresponding receiver, and a system, including a transmitter and the receiver, for wireless communication over a time-varying frequency-selective channel.

Description

METHOD OF RECEIVING A TRANSMITTED SIGNAL OVER A TIME- VARYING

FREQUENCY-SELECTIVE CHANNEL AND RECEIVER THEREOF

CROSS-REFERENCE TO RELATED APPLICATION

[0001] This application claims the benefit of priority of Singapore Patent Application No. 10202300195U filed on 25 January 2023, the content of which being hereby incorporated by reference in its entirety for all purposes.

TECHNICAL FIELD

[0002] The present invention generally relates to wireless communication over a time- varying frequency-selective channel, and more particularly, a method of receiving a transmitted signal over a time-varying frequency-selective channel, a receiver thereof, and a system, including a transmitter and the receiver, for wireless communication over a time-varying frequency-selective channel.

BACKGROUND

[0003] In mobile radio communication, radar sensing and satellite communication, very often time-varying channels are utilized in which the notorious Doppler shifts/spreads (frequency dispersiveness) are caused by moving transmitters, receivers or signal reflectors. Moreover, multipath propagation leads to high frequency selectivity (time dispersiveness). Thus, in general, practical wireless channels may be characterized as time-varying frequency- selective channels, which may also be referred to as doubly selective channels (DSCs). The high time and frequency dispersiveness of the DSC can significantly distort the transmitted signal, and thus efficient and accurate channel estimation and equalization techniques are desired.

[0004] For example, in high mobility systems with doubly selective (DS) fading, the fading time-variation may destroy the orthogonality among subcarriers and introduce inter-carrier interference (ICI), which may seriously degrade the performance of orthogonal frequency division multiplexing (OFDM) systems. To solve the channel estimation (CE) and ICI problem in OFDM, a lot of research has been carried out in the past decades For example, an iterative channel estimation and equalization technique has been proposed for OFDM signals transmitted over a DSC, which is modeled as a complex-exponential basis expansion model (CE-BEM). There has also been proposed a pilot-assisted time-varying channel estimation scheme for a DSC modeled by CE-BEM, and after channel estimation, a minimum mean square error (MMSE) equalizer is used for symbol detection. A message-passing (MP) equalizer (which has similar performance, but lower complexity than the MMSE equalizer) has also been proposed for detecting signals transmitted through a DSC. There has also been disclosed a structured distributed compressive sensing method for exploiting the sparsity of a DSC in the delay domain. Iterative channel estimation and equalization schemes have also been proposed for OFDM signals without cyclic prefix (CP) or for MIMO-OFDM signals transmitted over a DSC. [0005] Despite the large amount of research that has been carried out in the past, channel estimation and symbol detection are still challenging in a DSC because of high Doppler spread and a large number of multipaths that can significantly distort the signal transmitted over the DSC.

[0006] A need therefore exists to provide a wireless communication method over a time- varying frequency-selective channel, including a method of receiving a transmitted signal over a time-varying frequency-selective channel, that seek to overcome, or at least ameliorate, one or more of the deficiencies in conventional wireless communication methods over a time- varying frequency-selective channel, and more particularly, to improve the performance (e.g., the bit error rate (BER) performance) of channel estimation and symbol detection for the transmitted signal over the time-varying frequency-selective channel. It is against this background that the present invention has been developed.

SUMMARY

[0007] According to a first aspect of the present invention, there is provided a method of receiving a transmitted signal over a time-varying frequency-selective channel, the method comprising: obtaining a received symbol signal in frequency domain based on the transmitted signal; performing a pilot-aided channel estimation with respect to the time-varying frequency- selective channel using a first trained neural network model based on the received symbol signal to obtain a plurality of first estimated channel coefficients; performing a first equalization based on the received symbol signal and the plurality of first estimated channel coefficients to obtain a plurality of first detected source symbols; performing a data-aided channel estimation with respect to the time-varying frequency- selective channel using a second trained neural network model based on the received symbol signal and the plurality of first detected source symbols to obtain a plurality of second estimated channel coefficients; and performing a second equalization based on the plurality of second estimated channel coefficients to obtain a plurality of second detected source symbols.

[0008] According to a second aspect of the present invention, there is provided a receiver for receiving a transmitted signal over a time-varying frequency-selective channel, the receiver comprising: at least one memory; and at least one processor communicatively coupled to the at least one memory and configured to: obtain a received symbol signal in frequency domain based on the transmitted signal; perform a pilot-aided channel estimation with respect to the time-varying frequency- selective channel using a first trained neural network model based on the received symbol signal to obtain a plurality of first estimated channel coefficients; perform a first equalization based on the received symbol signal and the plurality of first estimated channel coefficients to obtain a plurality of first detected source symbols; perform a data-aided channel estimation with respect to the time-varying frequency- selective channel using a second trained neural network model based on the received symbol signal and the plurality of first detected source symbols to obtain a plurality of second estimated channel coefficients, and perform a second equalization based on the plurality of second estimated channel coefficients to obtain a plurality of second detected source symbols.

[0009] According to a third aspect of the present invention, there is provided a computer program product, embodied in one or more non-transitory computer-readable storage mediums, comprising instructions executable by at least one processor to perform a method of receiving a transmitted signal over a time-varying frequency-selective channel according to the above- mentioned first aspect of the present invention.

[0010] According to a fourth aspect of the present invention, there is provided a system for wireless communication over a time-varying frequency-selective channel, the system comprising: a transmitter configured to transmit a signal over the time-varying frequency-selective channel, and a receiver configured to receive the transmitted signal over the time-varying frequency- selective channel according to the above-mentioned second aspect of the present invention.

BRIEF DESCRIPTION OF THE DRAWINGS

[0011] Embodiments of the present invention will be better understood and readily apparent to one of ordinary skill in the art from the following written description, by way of example only, and in conjunction with the drawings, in which:

FIG. 1 depicts a schematic flow diagram of a method of receiving a transmitted signal over a time-varying frequency-selective channel, according to various embodiments of the present invention;

FIG. 2 depicts a schematic block diagram of a receiver for receiving a transmitted signal over a time-varying frequency-selective channel according to various embodiments of the present invention;

FIG. 3 depicts a schematic block diagram of an example mobile communication device in which the receiver for receiving a transmitted signal over a time-varying channel as described with reference to FIG. 2 may be embodied;

FIG. 4 depicts a system for wireless communication (which may also be referred to as a wireless communication system) over a time-varying frequency-selective channel, according to various embodiments of the present invention;

FIG. 5 depicts a schematic drawing of an example transmission block structure;

FIG. 6 depicts a schematic block diagram of OFDM transmission;

FIG. 7 depicts a schematic flow diagram of an example two-stage channel estimation and equalization method, according to various example embodiments of the present invention;

FIG. 8 depicts a schematic flow diagram of a pilot-aided channel estimation block, according to various example embodiments of the present invention;

FIG. 9 depicts a schematic flow diagram of the data-aided channel estimation block, according to various example embodiments of the present invention;

FIGs. 10 and 11 show example neural network parameter settings (Table 1) and example training parameter settings (Table II), respectively, used in the training of the first neural network model for pilot-added channel estimation and the second neural network model for data-aided channel estimation, according to various example embodiments of the present invention; FIG. 12 shows a table (Table III) showing a comparison of computational complexities of conventional channel estimation and equalization methods and the present channel estimation and equalization method (2S-BEM-DNN (2-stage basis expansion model deep neural network)) according to various example embodiments of the present invention;

FIG. 13 shows plots of computational complexities of different channel estimation and equalization techniques;

FIG. 14 shows a table (Table IV) showing example parameters of simulation conducted for the 2S-BEM-DNN method, according to various example embodiments of the present invention;

FIG. 15 shows plots of comparison of the normalized mean square errors (NMSE) performance for different channel estimation and equalization methods;

FIG. 16 shows plots of comparison of the BER performance for different channel estimation and equalization methods,

FIG. 17 shows plots of comparison of performance of the 2S-BEM-DNN method trained and tested under the same or different SNRs; and

FIG. 18 shows plots of comparison of performance of the 2S-BEM-DNN method trained and tested under the same or different Doppler.

DETAILED DESCRIPTION

[0012] Various embodiments of the present invention relate to wireless communication over a time-varying frequency-selective channel (which may be referred to as a doubly selective channel (DSC)), and more particularly, a method of receiving a transmitted signal over a time- varying frequency-selective channel, a receiver thereof, and a system, including a transmitter and the receiver, for wireless communication over a time-varying frequency-selective channel. [0013] As described in the background, in wireless communication over a time-varying frequency-selective channel, the high time and frequency dispersiveness of the time-varying frequency-selective channel can significantly distort the transmitted signal. In this regard, despite the large amount of research that has been carried out in the past, channel estimation and symbol detection are still challenging in a time-varying frequency-selective channel because of high Doppler spread and a large number of multipaths that can significantly distort the signal transmitted over the time-varying frequency-selective channel. Accordingly, various embodiments of the present invention provide a wireless communication method over a time- varying frequency-selective channel, including a method of receiving a transmitted signal over a time-varying frequency-selective channel, that seek to overcome, or at least ameliorate, one or more of the deficiencies in conventional wireless communication methods over a time- varying frequency-selective channel, and more particularly, to improve the performance (e.g., the bit error rate (BER) performance) of channel estimation and symbol detection for the transmitted signal over the time-varying frequency-selective channel.

[0014] FIG. 1 depicts a schematic flow diagram of a method 100 of receiving a transmitted signal over a time-varying frequency-selective channel (which may be referred to as a DSC), according to various embodiments of the present invention. The method 100 comprises: obtaining (at 106) a received symbol signal (e.g., a received symbol vector) in frequency domain based on the transmitted signal (received over the time-varying frequency-selective channel); performing (at 108) a pilot-aided channel estimation with respect to the time-varying frequency-selective channel using a first trained neural network model based on the received symbol signal to obtain a plurality of first estimated channel coefficients (e.g., a first estimated channel coefficient vector); performing (at 110) a first equalization based on the received symbol signal and the plurality of first estimated channel coefficients to obtain a plurality of first detected source symbols (e.g., a first detected source symbol vector); performing (at 112) a data-aided channel estimation with respect to the time-varying frequency-selective channel using a second trained neural network model based on the received symbol signal and the plurality of first detected source symbols to obtain a plurality of second estimated channel coefficients (e.g., a second estimated channel coefficient vector); and performing (at 114) a second equalization based on the plurality of second estimated channel coefficients to obtain a plurality of second detected source symbols (e.g., a second detected source symbol vector).

[0015] Accordingly, the method 100 of receiving a transmitted signal over a time-varying frequency-selective channel improves the performance (e g., the BER performance) of channel estimation and symbol detection for the transmitted signal over the time-varying frequency- selective channel. In particular, the pilot-aided channel estimation and the data-aided channel estimation are each performed using a respective trained neural network model according to various embodiments of the present invention (i.e., the first trained neural network model is trained to perform a pilot-aided channel estimation (i.e., trained to output (generate) the plurality of first estimated channel coefficients that correspond to a pilot-aided channel estimation) and the second trained neural network model is trained to perform a data-aided channel estimation (i.e., trained to output (generate) the plurality of second estimated channel coefficients that correspond to a data-aided channel estimation)), which have been found to significantly reduce the channel modelling error, thereby significantly improving the performance of channel estimation and symbol detection for the transmitted signal over the DSC. These advantages or technical effects will become more apparent to a person skilled in the art as the method 100 of receiving a transmitted signal, as well as the corresponding receiver, is described in more detail according to various embodiments and various example embodiments of the present invention.

[0016] In various embodiments, the time-varying frequency-selective channel is modeled based on a complex-exponential basis expansion model (CE-BEM). In this regard, the plurality of first estimated channel coefficients is a plurality of first estimated BEM coefficients and the plurality of second estimated channel coefficients is a plurality of second estimated BEM coefficients.

[0017] In various embodiments, the pilot-aided channel estimation is performed using the first trained neural network model based on a plurality of pilot symbols (e.g., a pilot vector) and a channel estimation matrix determined based on the received symbol signal.

[0018] In various embodiments, the first trained neural network model comprises: a first neural network portion configured to generate a plurality of initial first estimated channel coefficients (e g., an initial first estimated channel coefficient vector) in frequency domain based on the plurality of pilot symbols (e.g., the received pilot vector) and the channel estimation matrix; a channel coefficient converting layer configured to convert the plurality of initial first estimated channel coefficients in frequency domain into a plurality of initial first estimated channel coefficients in time domain; and a second neural network portion configured to generate a plurality of refined first estimated channel coefficients (e.g., a refined first estimated channel coefficient vector) in time domain, constituting the above-mentioned plurality of first estimated channel coefficients, based on the plurality of initial first estimated channel coefficients in time domain. Accordingly, in various embodiments, the first trained neural network model advantageously comprises a channel coefficient converting layer for converting the plurality of initial first estimated channel coefficients in frequency domain into time domain so as to enable the first trained neural network model to perform the pilot-aided channel estimation for a very fast time-varying channel (channel is changing within/over one data block instead of being constant within/over a data block). In particular, by converting the plurality of initial first estimated channel coefficients in frequency domain (constant over a data block) into time domain (time-varying over a data block), the first trained neural network model is advantageously able to perform the pilot-aided channel estimation for a very fast time- varying channel, thereby also enabling optimal symbol detection based on the pilot-aided channel estimation under such a fast fading channel. Accordingly, various embodiments advantageously address a technical problem of how to use a neural network model to estimate a very fast time-varying channel.

[0019] Tn various embodiments, the plurality of pilot symbols is comprised in a pilot vector. The first trained neural network model further comprises an input layer configured to convert the channel estimation matrix into a channel estimation one-dimensional (ID) array and concatenate the pilot vector and the channel estimation ID array to generate an input array to the first neural network portion (for performing the pilot-aided channel estimation).

[0020] In various embodiments, the first neural network portion is trained based on a first objective function configured to minimize an error (e.g., mean square error (MSE)) between the plurality of initial first estimated channel coefficients generated and a plurality of ground truth channel coefficients corresponding to the plurality of initial first estimated channel coefficients. In various embodiments, the second neural network portion is trained based on a second objective function configured to minimize an error (e g., MSE) between the plurality of refined first estimated channel coefficients generated and a plurality of ground truth channel coefficients corresponding to the plurality of refined first estimated channel coefficients.

[0021] In various embodiments, the second trained neural network model comprises: an input layer configured to generate a plurality of initial second estimated channel coefficients (e.g., an initial second estimated channel coefficient vector) in frequency domain based on the received symbol signal and the plurality of first detected source symbols; a channel coefficient converting layer configured to convert the plurality of initial second estimated channel coefficients in frequency domain into a plurality of initial second estimated channel coefficients in time domain; and a neural network portion configured to generate a plurality of refined second estimated channel coefficients (e.g., a refined second estimated channel coefficient vector) in time domain, constituting the above-mentioned plurality of second estimated channel coefficients, based on the plurality of initial second estimated channel coefficients in time domain. Accordingly, in various embodiments, the second trained neural network model advantageously comprises a channel coefficient converting layer for converting the plurality of initial second estimated channel coefficients in frequency domain into time domain so as to enable the second trained neural network model to perform the data-aided channel estimation for a very fast time-varying channel (channel is changing within/over one data block instead of being constant within/over a data block). In particular, by converting the plurality of initial second estimated channel coefficients in frequency domain (constant over a data block) into time domain (time-varying over a data block), the second trained neural network model is advantageously able to perform the data-aided channel estimation for a very fast time-varying channel, thereby also enabling optimal symbol detection based on the data-aided channel estimation under such a fast fading channel Accordingly, various embodiments advantageously address a technical problem of how to use a neural network model to estimate a very fast time- varying channel.

[0022] In various embodiments, the plurality of first detected source symbols comprises a plurality of detected data symbols and a plurality of detected pilot symbols. In this regard, the method 100 further comprises replacing the plurality of detected pilot symbols in the plurality of first detected source symbols with a plurality of known pilot symbols corresponding to the plurality of detected pilot symbols.

[0023] In various embodiments, the neural network portion of the second trained neural network model is trained based on a third objective function configured to minimize an error between the plurality of refined second estimated channel coefficients generated and a plurality of ground truth channel coefficients corresponding to the plurality of refined second estimated channel coefficients.

[0024] In various embodiments, the first equalization and the second equalization may be based on different types of equalization. In various embodiments, the first equalization is based on a banded minimum mean square error (BMMSE) equalization and the second equalization is based on a maximum-likelihood (ML) equalization. Although employing the BMMSE equalization and the ML equalization as the first equalization and the second equalization, respectively, may be preferred according to various embodiments (e g., they have been found to result in good or optimal computational efficiency and/or performance), it will be appreciated by a person skilled in the art that the present invention is not limited to any particular type of equalization for the first equalization and the second equalization, which may be selected or implemented as desired or as appropriate based on various factors.

[0025] In various embodiments, the first equalization is based on a BMMSE equalization. In this regard, the above-mentioned performing (110) the first equalization comprises: determining a first estimated channel matrix of the time-varying frequency-selective channel based on the plurality of first estimated channel coefficients; determining a banded first estimated channel matrix of the time-varying frequency -selective channel based on the first estimated channel matrix; and determining the plurality of first detected source symbols based on the received symbol signal and the banded first estimated channel matrix.

[0026] In various embodiments, the method 100 further comprises: determining a second estimated channel matrix of the time-varying frequency-selective channel based on the plurality of second estimated channel coefficients; performing an inter-carrier interference (ICT) removal with respect to the second estimated channel matrix to obtain an inter-carrier interference reduced (or removed) second estimated channel matrix; performing an inter-carrier interference removal with respect to the received symbol signal to obtain an inter-carrier interference reduced (or removed) symbol signal.

[0027] In various embodiments, the inter-carrier interference removal with respect to the received symbol signal is performed based on the received symbol signal, the plurality of first detected source symbols and the second estimated channel matrix.

[0028] In various embodiments, the second equalization is based on a maximum-likelihood (ML) equalization and is performed based on the inter-carrier interference reduced symbol signal and the inter-carrier interference reduced second estimated channel matrix.

[0029] In various embodiments, the above-mentioned obtaining (at 106) the received symbol signal comprises performing a discrete Fourier transform (DFT) based on the transmitted signal in time domain received to obtain the received symbol signal in frequency domain.

[0030] In various embodiments, the transmitted signal is transmitted over the time-varying frequency-selective channel based on orthogonal frequency division multiplexing (OFDM) transmission. Accordingly, in various embodiments, there is provided a method of transmitting a signal over the time-varying frequency-selective channel based on OFDM transmission.

[0031] In various embodiments, there is provided a wireless communication method comprising the above-mentioned method of transmitting a signal over a time-varying frequency-selective channel and the above-mentioned method 100 of receiving the transmitted signal over the time-varying frequency-selective channel as described herein with reference to FIG. 1 according to various embodiments of the present invention.

[0032] FIG. 2 depicts a schematic block diagram of a receiver 200 for receiving a transmitted signal over a time-varying frequency-selective channel according to various embodiments of the present invention, corresponding to the method 100 of receiving a transmitted signal over a time-varying frequency-selective channel as described herein according with reference to FIG 1 according to various embodiments of the present invention. The receiver 200 comprises: at least one memory 202; and at least one processor 204 communicatively coupled to the at least one memory 202 and configured to perform the method 100 of receiving a transmitted signal over a time-varying frequency-selective channel as described herein according to various embodiments of the present invention. Accordingly, the at least one processor 204 is configured to: obtain a received symbol signal in frequency domain based on the transmitted signal; perform a pilot-aided channel estimation with respect to the time-varying frequency-selective channel using a first trained neural network model based on the received symbol signal to obtain a plurality of first estimated channel coefficients; perform a first equalization based on the received symbol signal and the plurality of first estimated channel coefficients to obtain a plurality of first detected source symbols; perform a data-aided channel estimation with respect to the time-varying frequency-selective channel using a second trained neural network model based on the received symbol signal and the plurality of first detected source symbols to obtain a plurality of second estimated channel coefficients; and perform a second equalization based on the plurality of second estimated channel coefficients to obtain a plurality of second detected source symbols.

[0033] It will be appreciated by a person skilled in the art that the at least one processor 204 may be configured to perform various functions or operations through set(s) of instructions (e.g., software modules) executable by the at least one processor 204 to perform various functions or operations. Accordingly, as shown in FIG. 2, the system 200 may comprise: a symbol signal obtaining module (or a symbol signal obtaining circuit) 206 configured to obtain a received symbol signal in frequency domain based on the transmitted signal; a pilot-aided channel estimation module (or a pilot-aided channel estimation circuit) 208 configured to a perform a pilot-aided channel estimation with respect to the time-varying frequency-selective channel using a first trained neural network model based on the received symbol signal to obtain a plurality of first estimated channel coefficients; a first equalization module (or a first equalization circuit) 210 configured to perform a first equalization based on the received symbol signal and the plurality of first estimated channel coefficients to obtain a plurality of first detected source symbols; a data-aided channel estimation module (or a channel-aided channel estimation circuit) 212 configured to perform a data-aided channel estimation with respect to the time-varying frequency-selective channel using a second trained neural network model based on the received symbol signal and the plurality of first detected source symbols to obtain a plurality of second estimated channel coefficients; and a second equalization module (or a second equalization circuit) 214 configured to perform a second equalization based on the plurality of second estimated channel coefficients to obtain a plurality of second detected source symbols.

[0034] It will be appreciated by a person skilled in the art that the above-mentioned modules are not necessarily separate modules, and two or more modules of the system 200 may be realized by or implemented as one functional module (e.g., a circuit or a software program) as desired or as appropriate without deviating from the scope of the present invention. For example, two or more of the symbol signal obtaining module 206, the pilot-aided channel estimation module 208, the first equalization module 210, the data-aided channel estimation module 212 and the second equalization module 214 may be realized (e.g., compiled together) as one executable software program (e.g., software application or simply referred to as an “app”), which for example may be stored in the at least one memory 202 and executable by the at least one processor 204 to perform the corresponding functions or operations as described herein according to various embodiments.

[0035] In various embodiments, the receiver 200 for receiving a transmitted signal over a time-varying frequency-selective channel corresponds to the method 100 of receiving a transmitted signal over a time-varying frequency-selective channel as described herein with reference to FIG. 1, therefore, various operations, functions or steps configured to be performed by the least one processor 204 may correspond to various operations, functions or steps of the method 100 described herein according to various embodiments, and thus need not be repeated with respect to the receiver 200 for clarity and conciseness. In other words, various embodiments described herein in context of methods (e.g., the method 100 of receiving a transmitted signal over a time-varying frequency-selective channel) are analogously valid for the corresponding systems or devices (e.g., the system 200 for receiving a transmitted signal over a time-varying frequency-selective channel), and vice versa. For example, in various embodiments, the at least one memory 202 may have stored therein the symbol signal obtaining module 206, the pilot-aided channel estimation module 208, the first equalization module 210, the data-aided channel estimation module 212 and/or the second equalization module 214, which respectively correspond to various operations, functions or steps of the method 100 of receiving a transmitted signal over a time-varying frequency-selective channel as described herein according to various embodiments, which are executable by the at least one processor 204 to perform the corresponding operations, functions or steps.

[0036] A computing system, a controller, a microcontroller or any other system providing a processing capability may be provided according to various embodiments in the present invention. Such a system may be taken to include one or more processors and one or more computer-readable storage mediums. For example, the receiver 200 described herein may include at least one processor (or controller) 204 and at least one computer-readable storage medium (or memory) 202 which are for example used in various processing carried out therein as described herein. A memory or computer-readable storage medium used in various embodiments may be a volatile memory, for example a DRAM (Dynamic Random Access Memory) or a non-volatile memory, for example a PROM (Programmable Read Only Memory), an EPROM (Erasable PROM), EEPROM (Electrically Erasable PROM), or a flash memory, e.g., a floating gate memory, a charge trapping memory, an MRAM (Magnetoresistive Random Access Memory) or a PCRAM (Phase Change Random Access Memory).

[0037] In various embodiments, a “circuit” may be understood as any kind of a logic implementing entity, which may be special purpose circuitry or a processor executing software stored in a memory, firmware, or any combination thereof. Thus, in various embodiments, a “circuit” may be a hard-wired logic circuit or a programmable logic circuit such as a programmable processor, e.g., a microprocessor (e.g., a Complex Instruction Set Computer (CISC) processor or a Reduced Instruction Set Computer (RISC) processor). A “circuit” may also be a processor executing software, e.g., any kind of computer program, e g., a computer program using a virtual machine code, e.g., Java. Any other kind of implementation of various functions or operations may also be understood as a “circuit” in accordance with various embodiments. Similarly, a “module” may be a portion of a system according to various embodiments and may encompass a “circuit” as above, or may be understood to be any kind of a logic-implementing entity therefrom.

[0038] Some portions of the present disclosure may be explicitly or implicitly presented in terms of algorithms and functional or symbolic representations of operations on data within a computer memory. For example, these algorithmic descriptions and functional or symbolic representations are the means used by those skilled in the data processing arts to convey most effectively the substance of their work to others skilled in the art. It can be understood by a person skilled in the art that an algorithm is generally conceived to be a self-consistent sequence of steps leading to a desired result. For example, the steps require physical manipulations of physical quantities, such as electrical, magnetic or optical signals capable of being stored, transferred, combined, compared, and otherwise manipulated.

[0039] The present specification also discloses a system (e.g., which may also be embodied as one or more devices or apparatuses), such as the receiver 200, for performing various operations, functions or steps of various methods described herein. Such a system may be specially constructed for the required purposes or may comprise a general purpose computer system selectively activated or reconfigured by a computer program stored in the computer system. In general, various algorithms that may be presented herein are not limited to being implemented or executed by any particular computer system Alternatively, the construction of more specialized computer system to perform various operations, functions or steps of various methods described herein may be provided as desired or as appropriate without going beyond the scope of the present invention.

[0040] In addition, the present specification also at least implicitly discloses computer program(s) or software/functional module(s), in that it would be apparent to a person skilled in the art that various operations, functions or steps of various methods described herein may be put into effect by computer code The computer program(s) is not intended to be limited to any particular programming language and implementation thereof, and it will be appreciated by a person skilled in the art that a variety of programming languages and coding thereof may be used to implement the computer program(s). Moreover, the computer program(s) is not intended to be limited to any particular control flow as there are a variety of programming languages which can use different control flows. It will be appreciated by a person skilled in the art that a computer program may be stored on any computer-readable storage medium (non- transitory computer-readable storage medium), such as but not limited to, a magnetic disk, an optical disk or a memory chip. For example, a computer program stored on a computer-readable storage medium may be loaded and executed on a computer system to implement various operations, functions or steps of various methods described herein according to various embodiments of the present invention.

[0041] Accordingly, in various embodiments, there is provided a computer program product, embodied in one or more computer-readable storage mediums (non-transitory computer-readable storage medium), comprising instructions (e.g., the symbol signal obtaining module 206, the pilot-aided channel estimation module 208, the first equalization module 210, the data-aided channel estimation module 212 and/or the second equalization module 214) executable by one or more computer processors to perform a method 100 of receiving a transmitted signal over a time-varying frequency-selective channel as described hereinbefore with reference to FIG 1 according to various embodiments of the present invention. Accordingly, various computer programs or software modules described herein may be stored in a computer program product receivable by a system therein, such as the receiver 200 as shown in FIG. 2, for execution by at least one processor 204 of the system 200 to perform various operations, functions or steps of various methods described herein according to various embodiments of the present invention.

[0042] It will be appreciated by a person skilled in the art that various modules described herein (e.g., the symbol signal obtaining module 206, the pilot-aided channel estimation module 208, the first equalization module 210, the data-aided channel estimation module 212 and/or the second equalization module 214) may be software module(s) realized by computer program(s) or set(s) of instructions executable by a computer processor to perform various functions or operations. Various modules described herein (e.g., the symbol signal obtaining module 206, the pilot-aided channel estimation module 208, the first equalization module 210, the data-aided channel estimation module 212 and/or the second equalization module 214) may also be implemented as hardware module(s) being functional hardware unit(s) designed to perform various functions or operations. More particularly, in the hardware sense, a module is a functional hardware unit designed for use with other components or modules. For example, a module may be implemented using discrete electronic components, or it can form a portion of an entire electronic circuit such as an Application Specific Integrated Circuit (ASIC). Numerous other possibilities exist. It will also be appreciated by a person skilled in the art that a combination of hardware and software modules may be implemented. Furthermore, various operations, functions or steps of various methods described herein may be performed in parallel rather than sequentially as desired or as appropriate (e.g., as long as it does not render the method(s) inoperable or unsatisfactory for its intended purpose).

[0043] In various embodiments, the receiver 200 for receiving a transmitted signal over a time-varying frequency-selective channel may be realized by any computer system having communication functionality or capability (e g., a portable computer system, which may also be embodied as a computing device, such as a mobile communication device (e.g., a smartphone, a tablet computer, a wearable device)) including at least one processor and at least one memory. By way of an example only and without limitation, an example mobile communication device 300 is schematically shown in FIG. 3 in which the receiver 200 may be implemented. Various methods/steps or functional modules (e g., the symbol signal obtaining module 206, the pilot-aided channel estimation module 208, the first equalization module 210, the data-aided channel estimation module 212 and/or the second equalization module 214) may be implemented as software, such as a computer program being executed within the mobile communication device 300, and instructing the mobile communication device 300 (in particular, one or more processors therein) to perform various functions or operations as described herein according to various embodiments.

[0044] The example mobile communication device 300 may comprise a system unit 302, one or more input devices such as a keypad 304 and/or a touchscreen and one or more output devices such as a display screen 306. Tt will be appreciated by a person skilled in the art that the display screen 306 may be a touch-sensitive display screen, and thus may also constitute an input device (i.e., the display screen 306 may be an integrated input/ output device, and thus the keypad 304 may be omitted). For example, the system unit 302 may be coupled to a first communication unit 308 for wireless communication with a cellular network 310, for example, a 3G, 4G or 5G network or a future generation of cellular network. The system unit 302 may also be coupled to a second communication unit 312 for wireless communication with various communication networks 314, such as a local area network (LAN), a wireless personal area network (WPAN) or a wide area network (WAN). For example, the system unit 302 may include a processor 316, a Random Access Memory (RAM) 318 and a Read Only Memory (ROM) 320. The system unit 302 may also include a number of Input/Output (I/O) interfaces, for example, an I/O interface 322 to an output device and an I/O interface 324 to an input device. Various components of the system unit 302 may communicate via an interconnected bus 326 in a manner known to a person skilled in the art. For example, various software or application programs (or may simply be referred to herein as “apps”) may be pre-installed in a memory of the mobile communication device 300 or may be transferred (e.g., by reading from a memory card or by downloading wirelessly from a server) to a memory of the mobile communication device 300.

[0045] FIG. 4 depicts a system 400 for wireless communication (which may also be referred to as a wireless communication system) over a time-varying frequency-selective channel, according to various embodiments of the present invention. The system 400 comprises a transmitter 450 configured to transmit a signal over the time-varying frequency-selective channel and a receiver 200 configured to receive the transmitted signal (from the transmitter 450) over the time-varying frequency-selective channel as described herein with reference to FIG. 2 according to various embodiments of the present invention. In this regard, as described hereinbefore according to various embodiments, the transmitted signal is transmitted over the time-varying frequency-selective channel based on OFDM transmission.

[0046] It will be appreciated by a person skilled in the art that the terminology used herein is for the purpose of describing various embodiments only and is not intended to be limiting of the present invention. As used herein, the singular forms “a”, “an” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms “comprises” and/or “comprising,” when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof.

[0047] Any reference to an element or a feature herein using a designation such as “first”, “second” and so forth does not limit the quantity or order of such elements or features, unless stated or the context requires otherwise. For example, such designations may be used herein as a convenient way of distinguishing between two or more elements or instances of an element. Thus, a reference to first and second elements does not necessarily mean that only two elements can be employed, or that the first element must precede the second element, unless stated or the context requires otherwise. In addition, a phrase referring to “at least one of’ a list of items refers to any single item therein or any combination of two or more items therein.

[0048] In order that the present invention may be readily understood and put into practical effect, various example embodiments of the present invention will be described hereinafter by way of examples only and not limitations. It will be appreciated by a person skilled in the art that the present invention may, however, be embodied in various different forms or configurations and should not be construed as limited to the example embodiments set forth hereinafter. Rather, these example embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the present invention to those skilled in the art.

[0049] In particular, for better understanding of the present invention and without limitation or loss of generality, various example embodiments of the present invention will now be described with respect to wireless communication based on OFDM transmission over a time- varying frequency-selective channel (which may also be referred to as a doubly selective channel (DSC)), whereby the DSC is modeled as a complex-exponential basis expansion model (CE-BEM) and DFT (discrete fourier transform) bases are utilized for the CE-BEM. Furthermore, the wireless communication includes a method of receiving a transmitted signal over the DSC based on a two-stage channel estimation and equalization, whereby the first stage is based on a pilot-aided channel estimation and a banded minimum mean square error (BMMSE) equalization and the second stage is based on a data-aided channel estimation and a maxi mum -likelihood (ML) equalization. However, it will be understood by a person skilled in the art that the present invention is not limited to such an example implementation of the method of receiving a transmitted signal over the DSC and that various aspects of the example implementation may be varied or modified as desired or as appropriate without going beyond the scope of the present invention, as long as the pilot-aided channel estimation in the first stage and the data-aided channel estimation in the second stage are performed using a first trained neural network model (trained to generate an output (estimated channel coefficients) corresponding to a pilot-aided channel estimation) and a second trained neural network model (trained to generate an output (estimated channel coefficients) corresponding to a data-aided channel estimation), respectively, as described herein according to various example embodiments of the present invention For example, it will be understood by a person skilled in the art that various different bases may be used to model the channel, such as but not limited to, Fourier bases, DPS bases, polynomial bases and so on, as desired or as appropriate. Furthermore, it will be understood by a person skilled in the art that the present invention is not limited to the first equalization being BMMSE equalization and the second equalization being ML equalization, and different types of equalization for the first equalization and the second equalization may be employed as desired or as appropriate based on various factors. For example and without limitation, other types of equalization for the first and second stages may include MMSE or MP (matching pursuit) equalization. According to various example embodiments, the BMMSE equalization and the ML equalization are employed as the first equalization and the second equalization, respectively, as they have been found to result in good or optimal computational efficiency and/or performance.

[0050] Various example embodiments provide a method for wireless transmission over time-varying frequency-selective channels (also known as doubly selective channels (DSC)). Channel estimation and symbol detection are rather challenging in a DSC, especially for highly dispersive scenarios (high Doppler spread and large number of multipaths). In particular, channel estimation and symbol detection are challenging in a DSC because of high Doppler spread and a large number of multipaths that can significantly distort the signal transmitted over the DSC. In this regard, various example embodiments provide a wireless communication method over a DSC, including a method of receiving a transmitted signal over a DSC, that seek to overcome, or at least ameliorate, one or more of the deficiencies in conventional wireless communication methods over a DSC, and more particularly, to improve the performance (e g , the bit error rate (BER) performance) of channel estimation and symbol detection for the transmitted signal over the DSC. In order to reduce the number of channel estimation (CE) parameters, a complex-exponential basis expansion model (CE-BEM) has been proposed to model the DSC and is adopted to model the DSC according to various example embodiments of the present invention. In particular, to improve the performance of channel estimation and symbol detection for the transmitted signal over the DSC, various example embodiments train neural network models (e.g., deep neural network (DNN)) and employ the trained neural network models to estimate channel coefficients (e.g., CE-BEM channel coefficients) for OFDM transmission over the DSC, which has been found to significantly reduce the channel modelling error (e.g., the modelling error of CE-BEM), thereby improving (e.g., the BER performance) the performance of channel estimation and symbol detection for the transmitted signal over the DSC.

[0051] Based on the neural network aided (e.g., DNN-aided) channel estimation algorithms or techniques according to various example embodiments of the present invention, there is provided an example one-step two-stage channel estimation and equalization method for OFDM transmission over the DSC (which may be referred to herein as the 2S-BEM-DNN method (2-stage basis expansion model deep neural network) or simply as the present channel estimation and equalization method, e.g., corresponding to the method 100 of receiving a transmitted signal over a time-varying frequency-selective channel as described hereinbefore according to various embodiments of the present invention), which has been found to achieve superior channel estimation and symbol detection performance with low computational complexity. In various example embodiments, the 2S-BEM-DNN method is advantageously non-iterative, that is, only performs the two stages of channel estimation and equalization one time (no feedback loop) (e.g., in contrast to conventional iterative channel estimation and equalization methods which perform multiple iterations of channel estimation and equalization (i.e., include feedback loop)), and thus may be referred to as a one-step 2S-BEM-DNN method. For example, simulation results show that the 2S-BEM-DNN method according to various example embodiments significantly improves the BER performance of channel estimation and symbol detection for a transmitted signal over the DSC compared with conventional channel estimation and equalization techniques. In fact, it was surprisingly found that the BER performance of the 2S-BEM-DNN method approached very close to the BER lower bound even without performing any iteration of channel estimation and equalization. Moreover, the complexity of the 2S-BEM-DNN method according to various example embodiments grows only linearly with the OFDM frame length .V. [0052] Recently, deep learning (DL) and deep neural network (DNN) have drawn a lot of attention for its great success in various areas. Various example embodiments note that there are two key advantages of DL and DNN. Firstly, DL-based algorithms are data driven, and therefore are more robust to imperfections in real-world systems. Secondly, the execution of DNN can be highly parallelized on massively parallel processing architectures (e.g., graphic processing units (GPUs), specialized chips, etc.), which makes DL-based algorithms very efficient. To improve the performance of channel estimation and symbol detection for the transmitted signal over the DSC, various example embodiments introduce neural network models (e.g., DNN) to the physical layer and achieved superior performance in channel estimation and symbol detection in various practical applications. For example, various example embodiments address a technical problem of how to use a neural network model (e.g., DNN) to estimate a very fast time-varying channel (channel is changing within one data block) and how to make optimal symbol detection under such a fast fading channel. To address this technical problem, in various example embodiments, the neural network model for estimating channel coefficients is advantageously configured to comprise a channel coefficient converting layer for converting estimated channel coefficients (e g., pilot-aided or data-aided) in frequency domain into time domain so as to enable the neural network model to perform the channel estimation (e.g., pilot-aided or data-aided) for a very fast time-varying channel (channel is changing within/over one data block instead of being constant within/over a data block). Tn particular, by converting estimated channel coefficients in frequency domain (constant over a data block) into time domain (time-varying over a data block), the neural network model is advantageously able to perform the channel estimation for a very fast time-varying channel, thereby addressing the above-mentioned technical problem.

[0053] Accordingly, various example embodiments provide a one-step 2S-BEM-DNN method for OFDM transmission over a DSC. In various example embodiments, the DSC is modelled by a CE-BEM At a first stage of the detection of the transmitted signal received, the 2S-BEM-DNN method uses a first trained neural network model (e.g., DNN) to perform a pilot- aided channel estimation (CE) (i.e., the first trained neural network model is trained generate an output (estimated channel coefficients) that correspond to a pilot-aided channel estimation) and uses a banded-MMSE (BMMSE) equalizer for symbol detection. At a second stage, the 2S-BEM-DNN method uses a second trained neural network model (e.g., DNN) based on the detected symbols obtained from the first stage to perform a data-aided channel estimation (CE) (i.e., the second trained neural network model is trained generate an output (estimated channel coefficients) that correspond to a data-aided channel estimation) and uses an optimal maximum- likelihood (ML) equalizer, following an interference cancellation (IC) block, for symbol detection. Simulation results show that by using the 2S-BEM-DNN method according to various example embodiments, the channel estimation and BER performance can be significantly improved as compared to existing iterative channel estimation and equalization techniques, and furthermore, the complexity of the 2S-BEM-DNN method grows only linearly with the OFDM frame length N.

[0054] Accordingly, in various example embodiments, the first and second trained neural network models are advantageously employed to perform the functions of the pilot-aided and data-aided channel estimations, respectively, for a transmitted signal received over a DSC, which have been found to significantly improve the channel estimation performance over that achieved by conventional channel estimation techniques or algorithms. In this regard, in various example embodiments, a two-stage channel estimation and equalization method for OFDM transmission over a DSC has been provided based on a combination of neural network based channel estimations (e.g., DNN-based BEM channel estimations), hybrid equalization (e.g., different equalizers (e g., BMMSE and ML equalizers) in the first and second stages) and an inter-carrier interference (ICI) cancellation block to achieve joint channel estimation and equalization effectively (e.g., with improved BER performance).

System Model

[0055] For a better understanding, an example system model will now be described according to various example embodiments of the present invention. In the example system model, various example embodiments consider transmission over a DSC with one antenna each at the transmitter and the receiver. A data sequence x[n] of length N is transmitted at a rate of i symbols per second over the DSC. The discrete-time baseband equivalent of the received symbol at the n^th time instant may be expressed as: where h[n; Z] denotes the discrete-time equivalent baseband representation of the DSC, which subsumes the physical multipath channel together with the transmit and receive pulse shaping filters, Z denotes the I^th multipath, L denotes the number of multipaths and may be given as L — with T_max being the maximum delay spread of the channel, and v[n] denotes the circularly symmetric complex additive white Gaussian noise (AWGN) with v[n]~CW(0, crj). [0056] In various example embodiments, an example block transmission design may be adopted where, in frequency-domain, the pilot tones (or pilot symbols) are multiplexed with the data subcarriers (or data symbols) periodically, such as illustrated in FIG. 5. In particular, FIG. 5 depicts a schematic drawing of an example transmission block structure. It will be appreciated by a person skilied in the art that such periodic pilot placement is also applicable to, for example, LTE (Long-Term Evolution) with some easy modifications. As shown in FIG. 5, the transmission block may comprise three sub-blocks, each sub-block comprising a data sub-block (represented by filled/solid circles in FIG. 5) and a pilot sub-block (which may also be referred to as a pilot cluster) (represented by hollow circles in FIG. 5). Each pilot sub-block includes a pilot tone at the center thereof and surrounded by Q null subcarriers on both sides of the center, where Q denotes the number of channel coefficients (e.g., BEM coefficients) which will be further described later below. The placement of null subcarriers on both sides of the pilot tone is beneficial in fast fading, because otherwise, the Doppler spread may introduce interference between pilot and data, thereby distorting the OFDM data and channel estimate. In various example embodiments, the pilot positions may be optimized to further improve the channel estimate performance, at the expense of increasing computational complexity.

[0057] In various example embodiments, the pilot structure adopted belongs to the frequency domain Kronecker delta (FDKD) family, which has been widely used for frequency domain pilot designs and thus need not be described in detail herein. For FDKD pilots, the pilot sub-carriers are grouped as clusters, with one active pilot surrounded by guard (null) sub- carriers, which facilitate the removal or reduction of inter-carrier interference (ICI) as will be described later below according to various example embodiments.

Channel Model

[0058] For a better understanding, an example channel model will now be described according to various example embodiments of the present invention. In various example embodiments, the channel h[n; /] (e.g., corresponding to the time-varying frequency-selective channel (DSC) as described hereinbefore according to various embodiments of the present invention) may be modeled using the CE-BEM, where the I^th tap of the channel (I^th channel tap, which may also be referred to as I^th path) at the n^th time-instant is expressed as a weighted combination of the complex exponentials bases functions. In this regard, it will be appreciated by a person skilled in the art that various different bases may be used to model the channel, such as but not limited to, Fourier bases, DPS bases, polynomial bases and so on, as desired or as appropriate. As an illustrative example, various example embodiments show that optimizing the DFT bases for CE-BEM can significantly improve the channel estimation significantly under various fading rates. For example, the channel may be expressed as:

(Equation 2) where a>_q [ ]) } denotes the BEM modeling frequency, K ≥ N is the BEM resolution, ) denotes the weight or the q^th BEM coefficient corresponding to the I^th path, and Q denotes the number of BEM coefficients. In various example embodiments, Q may be given as where f_max is the channel maximum Doppler spread. In this regard, to fully remove the inter-carrier interference, the number of null subcarriers on each side of the active pilot of a pilot cluster needs to be equal to or larger than 0. Accordingly, to achieve higher transmission efficiency, Q null subcarriers are implemented according to various example embodiments of the present invention. In various example embodiments, the BEM modeling frequency is taken to be uniformly distributed between In various other example embodiments, non-uniformly spaced frequencies may be adopted for the modeling frequency to further reduce the BEM channel modelling error.

[0059] Accordingly, by substituting Equation (2) into Equation (1), the received symbol signal at the n^th time instant may be expressed as:

(Equation 3) |0060| Defining y — [y[0], ...,y[N — 1]] , in a block-level, the received symbol signal (or the received symbol vector) y may be expressed as:

(Equation 4) where D_q is the diagonal matrix with the q^tl1 basis function components on its diagonal, i.e., , Z_t is an N X (A + L) Toeplitz matrix defined as N X N identity matrix. The transmitted symbol vector x is defined , and v is the additive noise vector defined as v =

[0061] FIG. 6 depicts a schematic block diagram of OFDM transmission. For OFDM transmission, the frequency-domain information symbols may firstly be divided into transmission blocks of N symbols, with each symbol transmitted on a subcarrier in parallel. Each transmission block may then be transformed to time domain by inverse discrete Fourier transform (IDFT). Then, a cyclic prefix (CP) of length v > L may be added in front of each transmission block For example, the CP may actually be the repeat of the last v points of each transmission block after IDFT. The CP-added transmission blocks may then be transmitted through the channel. Suppose S [fc] is the frequency-domain symbol which is transmitted by the k ^,h subcarrier, after IDFT, the n^th symbol in time-domain ( x[n] ) transmitted (e g , corresponding to the transmitted signal in the method 100 as described hereinbefore according to various embodiments) may be expressed as:

(Equation 5)

[0062] At the receiver side, after removing the CP, the block-level received symbol signal

(or the block-level received symbol vector) y in time-domain may be expressed as:

(Equation 6) where F is the N X N unitary DFT matrix with the element on the i^th row and j^th column equals to denotes Hermitian transpose. V circular matrix with the first column having 1 in position I + 1, that is:

(Equation 7) T and S = [S[0] S[1V - 1]] is the source symbol vector. After DFT, the block-level received symbol vector Y in frequency domain (e.g., corresponding to the received symbol signal in frequency domain in the method 100 as described hereinbefore according to various embodiments) may be expressed as:

(Equation 8) wher is the noise vector in the frequency-domain, with F[Zc] being the noise on the k^th subcarrier.

Two-Stage Channel Estimation and Equalization Method

[0063] FIG. 7 depicts a schematic flow diagram of an example two-stage channel estimation (CE) and equalization method 700 (2S-BEM-DNN) according to various example embodiments of the present invention (e.g., corresponding to the method 100 of receiving a transmitted signal over a time-varying frequency-selective channel as described hereinbefore according to various embodiments of the present invention).

[0064] The method 700 comprises: obtaining a received symbol signal Y (e.g., a received symbol vector) in frequency domain based on the transmitted signal (e g., after the received symbol signal in time domain has been converted by the DFT into frequency domain as shown in FIG. 6); performing (e.g., by a pilot-aided channel estimation module or block 708) a pilot- aided channel estimation with respect to the DSC using a first trained neural network model (e.g., DNN) based on the received symbol signal Y to obtain a plurality of first (e.g., pilot-aided) estimated channel coefficients (e.g., a first estimated channel coefficient vector, e.g., comprising first estimated BEM coefficients) /r^p)[n; Z]; performing (e.g., by a first equalization module or block 710) a first equalization (e g., banded minimum mean square error (BMMSE) equalization) based on the received symbol signal Y and the plurality of first estimated channel coefficients h^[n; Z] to obtain a plurality of first (e.g., pilot-aided) detected source symbols (e.g., a first detected source symbol vector S'); performing (e.g., by a data-aided channel estimation module or block 712) a data-aided channel estimation with respect to the DSC using a second trained neural network model (e.g., DNN) based on the received symbol signal Y and the plurality of first detected source symbols S to obtain a plurality of second (e.g., data-aided) estimated channel coefficients (e.g., a second estimated channel coefficient vector, e.g., comprising second estimated BEM coefficients) h^(d) [n; Z]; and performing (e g , by a second equalization module or block 714) a second equalization (e g., maximum-likelihood (ML) equalization) based on the plurality of second estimated channel coefficients fi^(d! [n; Z] to obtain a plurality of second (e.g., data-aided) detected source symbols (e g., a second detected source symbol vector S) [0065] As an illustrative example, as shown in FIG. 7, at the first stage, the received signal vector F, together with the known pilot symbols, is firstly used by the first trained neural network model to perform (at 708) a pilot-aided channel estimation to output a first (e.g., pilot- aided) estimated channel coefficient vector [n; Z] . Subsequently, the first estimated channel coefficient vector h'^p)[n; Z] is used by a banded-MMSE equalizer to perform (at 710) data/symbol detection in frequency domain to obtain a first (e.g., pilot-aided) detected symbol vector S At the second stage, the first detected symbol vector S output from the banded-MMSE equalizer and the received signal vector Y are be used by the second trained neural network model to perform (at 712) data-aided channel estimation to obtain a second (e g , data-aided) estimated channel coefficient vector ft® [n; Z], which is then used by a ML equalizer to perform (at 714) data/symbol detection to obtain a second or final (e.g., data-aided) detected symbol vector S. In various example embodiments, the first detected source symbol vector S comprises a plurality of detected data symbols and a plurality of detected pilot symbols. In this regard, the plurality of detected pilot symbols in the first detected source symbol vector S is replaced with a plurality of known pilot symbols corresponding to the plurality of detected pilot symbols. For example, such a replacement may be performed by the first equalization block 710.

[0066] Each functional block shown in FIG. 7 will now be described below in further detail according to various example embodiments of the present invention.

Pilol-aided Channel Estimation (CE)

[0067] FIG. 8 depicts a schematic flow diagram of the pilot-aided channel estimation (CE) block 708, comprising the first trained neural network model, according to various example embodiments of the present invention. As shown in FIG. 8, the pilot-aided channel estimation is performed using the first trained neural network model based on a plurality of pilot symbols (e.g., a pilot vector) y^> and a channel estimation matrix W (which may also be referred to as an interpolation matrix) (e.g., MMSE estimation matrix IF_LMMSE, which may also be referred to as an MMSE interpolation matrix) determined based on the received symbol signal Y. . The plurality of pilot symbols (e.g., a pilot vector) and the channel estimation matrix W may be determined in a manner known in the art, such as but not limited to the manner as described in Tang el al., “Pilot-assisted time-varying channel estimation for ofdm systems,” IEEE Trans. Signal Process., vol. 55, no. 5, pp. 2226-2238, May 2007. As shown in FIG. 8, the first trained neural network model comprises: a first neural network portion 810 configured to generate a plurality of initial first (pilot-aided) estimated channel coefficients (e g , an initial first (pilot- aided) estimated channel coefficient vector, e.g., comprising initial first (pilot-aided) estimated BEM coefficients) c^fp> in frequency domain based on the pilot vector y^(p> and the channel estimation matrix 1V_LMMSE; a channel coefficient converting layer (e.g., a BEM layer) 812 configured to convert the plurality of initial first (pilot-aided) estimated channel coefficients in frequency domain c^(p> into a plurality of initial first (pilot-aided) estimated channel coefficients in time domain Z]; and a second neural network portion 820 configured to generate a plurality of refined first (pilot-aided) estimated channel coefficients in time domain h^[n; Z], constituting the above-mentioned plurality of first estimated channel coefficients, based on the plurality of initial first (pilot-aided) estimated channel coefficients in time domain Z], [0068] In various example embodiments, as shown in FIG 8, the first trained neural network model may comprise an input layer 808, an output layer 824, and a number of inner layers (e g., fully connected layers). In various example embodiments, the inner layers are divided into two parts or portions, namely, inner layer(s) for initial estimation in the frequency domain (comprised in the first neural network portion 810) and inner layer(s) for refined estimation in the time domain (comprised in the second neural network portion 820), with a BEM layer 812 between them. Based on such a configuration of the first trained neural network model, the first and second neural network portions 810, 820 are configured to estimate channel coefficients in the frequency and time domains, respectively, whereby the plurality of pilot- aided estimated channel coefficients in frequency domain c'^p> is advantageously converted by the BEM layer 812 into a plurality of pilot-aided estimated channel coefficients in time domain Z] so as to enable the first trained neural network model to perform the pilot-aided channel estimation for a very fast time-varying channel (channel is changing within/over one data block instead of being constant within/over a data block).

[0069] The computation of the j¹¹¹ neuron at each inner layer may be given by:

(Equation 9) where bj, Oj, c_t, and W_tJ and I are the bias, output data, input data, weights and the number of neurons, respectively /(•) is a non-linear function, called an activation function For example, the non-linear function may be a Sigmoid function or a Relu function, which may be defined respectively.

[0070] In various example embodiments, inputs of the pilot-aided channel estimation block 708 may include the MMSE estimation matrix _anc| _th_e received pilot vector whereby and denotes the i^th received pilot cluster. In various example embodiments, the first trained neural network model further comprises an input layer 808 configured to convert the channel estimation matrix WLMMSE '^{nt0 a} ID array and concatenate the received pilot vector y'^p! and the channel estimation ID array to generate an input array as input to the first neural network portion 810 (for performing the pilot-aided channel estimation). The input array may then be processed by the first neural network portion 810 (e g., corresponding to the initial estimation part or portion of the first neural network model), which outputs an initial estimated channel coefficient vector (e.g., comprising initial estimated BEM coefficients in frequency domain) . The initial estimated channel coefficient vector in frequency domain may then be input to the BEM layer 812 which is configured to convert the initial estimated channel coefficient vector in frequency domain c'^p! into an initial estimated channel coefficient vector in time domain h^{(p l}[n; Z], for example, as follows:

(Equation 10) [0071] The initial estimated channel coefficient vector in time domain h'^:p) [n; Z] may then be input to the second neural network portion 820 (e.g., corresponding to the refined estimation part or portion of the first neural network model), which is configured to output the refined estimated channel coefficient vector in time domain Z] (e g., as final output of the pilot- aided channel estimation block 708). Accordingly, in various example embodiments and for example, since the first trained neural network model uses the pilot vector y^ together with known pilot to perform channel estimation for the pilot subcarriers and uses the channel estimation matrix W to extend the channel estimation to other subcarriers, the first trained neural network model performs a pilot-aided channel estimation.

[0072] In an example implementation shown in FIG. 8, the first and second neural network portions 810, 820 each comprises fully connected layers as inner layers. The function of the first and second neural network portions 810, 820 may each be represented by y = /(x), where x and y denote the input and the output of the neural network portion, respectively, and f () denotes the neural network function. The neural network function is associated with various parameters (e g , weights and bias), and for example, Equation (9) above shows various parameters associated with the j*¹¹ neuron at each inner layer. In this regard, by minimizing a loss function (e.g., Equation (20) for the first neural network portion 810 and Equation (21) for the second neural network portion 820, which will be described later below), parameters associated with the neural network function are optimized during training. It will be appreciated by a person skilled in the art that the first and second neural network portions 810, 820 are each not limited to being implemented by fully connected layers, and may be implemented by other neural network configuration or structure as desired or as appropriate (such as but not limited to attention block) as long as the first neural network portion 810 is capable of being trained to generate the plurality of initial estimated channel coefficients (e g., the initial estimated channel coefficient vector) in frequency domain and the second neural network portion 820 is capable of being trained to generate the plurality of refined first (pilot-aided) estimated channel coefficients in time domain as described herein according to various example embodiments.

First (e.g., Banded-MMSE) Equalization

[0073] After the pilot-aided channel estimation, a first (e g., banded-MMSE (BMMSE)) equalization may be performed for symbol detection. In various example embodiments, the first equalization may comprise: determining a first (e g., pilot-aided) estimated channel matrix of the DSC based on the plurality of first (e.g., pilot-aided) estimated channel coefficients I], determining a banded first estimated channel matrix H_K of the DSC based on the first estimated channel matrix and determining the plurality of first (e.g., pilot-aided) detected source symbols S based on the received symbol signal Y and the banded first estimated channel matrix H_K.

100741 As an illustrative example, an example BMMSE equalization method or algorithm will now be described according to various example embodiments of the present invention. The example BMMSE equalization method may comprise:

1) Determine/estimate/reconstruct the original channel matrix H^(p> (corresponding to the first estimated channel matrix) by:

2) Determine matrix G by:

(Equation 12) where is the banded approximation of the original channel matrix with being an N X N matrix whose main diagonal, K sub-diagonals, and K super-diagonals are ones, and the remaining entries are zero. O denotes element-wise multiplication. R_n is the covariance matrix of the noise, and regularization term which serves to prevent/minimise performance degradation at high SNR.

3) Determine the detected source symbol vector (e.g., corresponding to the first (e.g., pilot-aided) detected source symbol vector) by:

(Equation 13) Data-aided Channel Estimation (CE)

[0075] FIG. 9 depicts a schematic flow diagram of the data-aided channel estimation (CE) block 712, comprising the second trained neural network model, according to various example embodiments of the present invention. As shown in FIG. 9, the data-aided channel estimation is performed using the second trained neural network model based on the received symbol signal Y and the plurality of first detected source symbols S to obtain a plurality of second (data- aided) estimated channel coefficients [n; Z] (e.g., a second (data-aided) estimated channel coefficient vector) In particular, as shown in FIG 9, the second trained neural network model comprises an input layer 908 configured to generate a plurality of initial second (data-aided) estimated channel coefficients (e g., an initial second (data-aided) estimated channel coefficient vector, e.g., comprising initial second (data-aided) estimated BEM coefficients) in frequency domain c® based on the received symbol signal Y and the plurality of first (pilot-aided) detected source symbols S, a channel coefficient converting layer (e.g., a BEM layer) 912 configured to convert the plurality of initial second (data-aided) estimated channel coefficients in frequency domain c® into a plurality of initial second (data-aided) estimated channel coefficients in time domain h^^d) [n; Z] ; and a neural network portion 920 configured to generate a plurality of refined second (data-aided) estimated channel coefficients (e.g., a refined second (data-aided) estimated channel coefficient vector) in time domain H^(d)[n; Z], constituting the above-mentioned plurality of second (data-aided) estimated channel coefficients, based on the plurality of initial second (data-aided) estimated channel coefficients in time domain /i^ [n; Z], [0076] In various example embodiments, as shown in FIG. 9, the second trained neural network model may comprise the input layer 908, an output layer 924 and a number of inner layers (e.g., fully connected layers). The neural network portion 920 may comprise the inner layers. Inputs of the data-aided channel estimation block 712 may include the first (pilot-aided) detected source symbol vector S and the received symbol vector Y As described hereinbefore, the first detected source symbol vector S may comprise a plurality of detected data symbols and a plurality of detected pilot symbols (which also be referred to as data sub-blocks and pilot sub- blocks (or clusters)) In this regard, as the pilot sub-blocks are known to the receiver, the pilot sub-blocks in S may be replaced by the known pilot sub-blocks, and thus, the first detected source symbol vector S input to the data-aided channel estimation block 712 may be referred to as the pseudo pilot comprising both the known pilot symbols and the recovered/ detected data symbols. As shown in FIG. 9, the input layer 908 may be configured to concatenate the first detected source symbol vector S and the received symbol vector Y to generate an input array (for generating the above-mentioned plurality of initial second (data-aided) estimated channel coefficients in frequency domain c*®).

[0077] According to Equation (8), the received symbol signal Y may be expressed as the true BEM coefficient vector. R i_{s a} matrix with the ((Q + 1)/ + q)^th column equals to

FD_qZ_lF^HS A least square (LS) estimation of the true BEM coefficient vector may be given by Equation (14) below. As S is unknown, S may be used to replace S in R Accordingly, in various example embodiments, the input layer 908 may be configured to convert the inputs (the above-mentioned input array) into the estimated BEM coefficients as follows: (Equation 14) where i_{s a} matrix with the ((<? column equ als to [0078] The estimated BEM coefficients may then be input to the BEM layer 912, which is configured to convert estimated BEM coefficients in frequency domain c^(d! into the time domain estimated BEM coefficients [n; Z], for example, as follows:

(Equation 15) The time domain estimated BEM coefficients may then be input to the neural network portion 920 of the neural network model, which outputs the refined estimated time-domain channel coefficients h^[n; /] (e g , as final output of the data-aided channel estimation block 712). Based on such a configuration of the second trained neural network model, the input layer 908 and the neural network portion 920 are configured to estimate channel coefficients in the frequency and time domains, respectively, whereby the plurality of data-aided estimated channel coefficients in frequency domain is advantageously converted by the BEM layer 912 into a plurality of data-aided estimated channel coefficients in time domain h^(d)\n; I] so as to enable the second trained neural network model to perform the data-aided channel estimation for a very fast time-varying channel (channel is changing within/over one data block instead of being constant within/over a data block).

[0079] Accordingly, in various example embodiments and for example, since the second trained neural network model uses the received symbol signal Y and the detected symbol vector S (pseudo pilot) to perform channel estimation for all the subcarrier, including the pilot and non-pilot subcarriers, the second trained neural network model performs a data-aided channel estimation.

[0080] In an example implementation shown in FIG. 9, the neural network portion 920 comprises fully connected layers as inner layers. The function of the neural network portion 920 may be represented by y — f(x), where x and y denote the input and the output of the neural network portion 920, respectively, and f () denotes the neural network function. The neural network function is associated with various parameters (e g., weights and bias), and for example, Equation (9) above shows various parameters associated with the y"¹ neuron at each inner layer. In this regard, by minimizing a loss function (e.g., Equation (22) which will be described later below), parameters associated with the neural network function are optimized during training. It will be appreciated by a person skilled in the art that the neural network portion 920 is not limited to being implemented by fully connected layers, and may be implemented by other neural network configuration or structure as desired or as appropriate (such as but not limited to attention block) as long as it is capable of being trained to generate the plurality of refined second (data-aided) estimated channel coefficients (e g., a refined second (data-aided) estimated channel coefficient vector) in time domain /] based on the plurality of initial second (data-aided) estimated channel coefficients in time domain h^[n; Z] as described herein according to various example embodiments. Interference Cancellation (IC)

[0081] In various example embodiments, inter-carrier interference (ICI) removal may be performed by the ICI cancellation module or block 720. In this regard, a second (data-aided) estimated channel matrix of the DSC may be determined based on the plurality of second

(data-aided) estimated channel coefficients (e.g., the second estimated channel coefficient vector) in time domain h^fn; Z], An inter-carrier interference removal may then be performed with respect to the second estimated channel matrix H^(d) to obtain an inter-carrier interference reduced (or removed) second estimated channel matrix H In addition, an inter-carrier interference removal may be performed with respect to the received symbol signal Y to obtain an inter-carrier interference reduced symbol signal Y' . In this regard, the inter-carrier interference removal with respect to the received symbol signal Y may be performed based on the received symbol signal Y, the plurality of first (pilot-aided) detected source symbols 5 and the second estimated channel matrix H^(d>.

[0082] As an example, the ICI cancellation block 720 may be configured to remove the intercarrier interference from the sub-diagonals and super-diagonals of the channel matrix (except the main B_Hb sub- and super-diagonals) of the DSC. After the ICI removal, the channel matrix becomes a banded channel matrix H with dispersion width equal to Q + 1. For example, according to the OFDM concept, the received symbol signal Y is given by Y = HfS + V = . Hf is the frequency domain channel matrix which is given by . In reality, as h_t is unknown, the data-aided estimated channel matrix may be used to replace h_t in the computation of the frequency domain channel matrix. In particular, before the ICI removal, the data-aided estimated frequency domain channel matrix H^(d> (i.e., the channel matrix obtained based on the estimated channel coefficients h^<d)[n; Z] generated from the data-aided channel estimation) may be given by:

(Equation 16) After the ICI removal, the banded channel matrix being an matrix whose main diagonal, sub-diagonals, and super-diagonals are ones, and the remaining entries are zero O denotes element-wise multiplication. [0083] After the ICI removal, according to various example embodiments, the ICI reduced (or removed) symbol signal Y' may be obtained based on the received symbol signal Y as follows:

(Equation 17) where

Second (e.g., ML) Equalization

[0084] After the data-aided channel estimation, a second (e.g., maximum-likelihood (ML)) equalization may be performed for symbol detection. In various example embodiments, the second equalization is performed based on the ICI reduced symbol signal Y' and the banded (i.e., the inter-carrier interference reduced) second estimated channel matrix H.

[0085] As an illustrative example, an example ML equalization method or algorithm will now be described according to various example embodiments of the present invention. The example ML equalization method may comprise:

1) Start from the 0^th subcarrier. For each state s₀(i), i E {0,1, — 1} , initialize the accumulated metric as r(s₀(i)) — 0.

2) For each Sk-itjm), ™ 6 {0,1, ..., M — 1}, compute the branch metric by:

(Equation 18) where F'[fc] is the k^th element in is the element on the

I th q\ column of ft | ■ | _N denotes modulo N.

IN

3) Determine the accumulated metric of the k^th subcarrier by:

4) Keep the preceding state j_m which leads to minimum T )) in a table. and go back to step 2 or go to step

6) Find the most likely transmitted symbol sequence S recursively by tracing back from the state with minimum accumulated metric [0086] Accordingly, the two-stage channel estimation and equalization (2S-BEM-DNN) method 700 (or corresponding algorithm) for OFDM transmission according to various example embodiments of the present invention may include:

1) Perform pilot-aided channel estimation using the pilot-aided channel estimation block 708 (including the first trained neural network model) described hereinbefore with reference to FIG. 8 according to various example embodiments of the present invention. For example, the pilot- aided estimated channel coefficients are /] and the reconstructed pilot-aided channel matrix H^(p) is derived according to Equation (11).

2) Using the BMMSE equalizer described hereinbefore according to various example embodiments to perform the first stage of equalization.

3) Pass the equalized symbol vector S to the data-aided channel estimation block 712. Perform data-aided channel estimation using the data-aided channel estimation block 712 (including the second trained neural network model) described hereinbefore with reference to FIG. 9 according to various example embodiments of the present invention. For example, the data- aided estimated channel coefficients are [n; Z] and the reconstructed data-aided channel matrix H^^d) is derived according to Equation (16)

4) Perform ICI removal on the data-aided estimated channel matrix H^<d>, and the data-aided estimated channel matrix after the ICI removal is H. The received symbol signal Y' after ICI removal may be obtained according to Equation (17).

5) Using the ML equalizer described hereinbefore according to various example embodiments to perform the second stage of equalization and make a final decision.

Neural Network Training

[0087] For the first neural network model for the pilot-aided channel estimation, as it comprises a first neural network portion 810 and a second neural network portion 820, according to various example embodiments, the training of the first neural network model may also be divided into two stages or steps. In the first stage, parameters of the first neural network portion 810 (e.g., inner layers thereof) for the pilot-aided initial estimation are trained, while parameters of the second neural network portion 820 (e g., inner layers thereof) for the pilot- aided refined estimation are fixed. In this regard, the first neural network portion 810 may be trained based on a first objective function configured to minimize an error between the plurality of initial first (pilot-aided) estimated channel coefficients in frequency domain generated and a plurality of ground truth channel coefficients in frequency domain c corresponding to the plurality of initial first (pilot-aided) estimated channel coefficients in frequency domain

For example, the first objective function may be configured to minimize the mean square error (MSE) between the pilot-aided channel coefficients in frequency domain c'^p! (e.g., BEM coefficients) estimated by using pilots and the true channel coefficients in frequency domain c (e g., BEM coefficients), for example, as follows:

(Equation 20) where || ■ ||₂ denotes 2-norm. For example, the training data may be obtained using Matlab to generate multipath channels with known channel coefficients.

[0088] Next, in the second stage, the parameters of the second neural network portion 820 (e.g., inner layers thereof) for the pilot-aided refined estimation are trained, while the parameters of the first neural network portion 810 (e g , inner layers thereof) for the pilot-aided initial estimation are fixed. In this regard, the second neural network portion 820 is trained based on a second objective function configured to minimize an error between the plurality of refined first (pilot-aided) estimated channel coefficients in time domain (nJ) generated and a plurality of ground truth channel coefficients in time domain h(n, I) corresponding to the plurality of refined first (pilot-aided) estimated channel coefficients H^(p)(n, Z) . For example, the second objection function may be configured to minimize the MSE between the pilot-aided estimated time domain channel coefficients h^fnj] and the true time domain channel coefficients Zi[n; Z] (n — 0, 1, ..., N — 1), for example, as follows:

(Equation 21) Similarly, for example, the training data may be obtained using Matlab to generate multipath channels with known channel coefficients.

[0089] Similarly, the neural network portion 920 of the second trained neural network model may be trained based on a third objective function configured to minimize an error between the plurality of refined second (data-aided) estimated channel coefficients in time domain h^(a)(n, Z) generated and a plurality of ground truth channel coefficients in time domain h(n, Z) corresponding to the plurality of refined second (data-aided) estimated channel coefficients in time domain h^fn, Z). For example, the third objective function for the data- aided channel estimation may be configured to minimize the MSE between the data-aided estimated time domain channel coefficients [n; I] and the true time domain channel coefficients h[n; Z] (n = 0, 1, ..., N — 1), for example, as follows:

(Equation 22) Similarly, for example, the training data may be obtained using Matlab to generate multipath channels with known channel coefficients.

[0090] For illustrative purposes and by way of examples only, FIGs. 10 and 11 show example neural network parameter settings (Table I) and example training parameter settings (Table II), respectively, used in the training of the first and second neural network models, according to various example embodiments of the present invention. For example, as can be seen in Table I, the BEM layer 812, 912 does not contain neurons.

|0091| Various advantages and improvements of the present channel estimation and equalization method over DSC according to various example embodiments of the present invention over conventional channel estimation and equalization methods will now be described.

Improvement on Computational Complexity

[0092] The computational complexities of conventional channel estimation and equalization methods (the iterative channel estimation and MLSE proposed by Barhumi et al., “MLSE and MAP equalization for transmission over doubly selective channels,” IEEE Trans. Veh. Technol., vol. 58, no. 8, pp. 4120-4128, Oct. 2009, the pilot-aided channel estimation and MMSE equalization proposed by Tang et al., “Pilot-assisted time-varying channel estimation for ofdm systems,” IEEE Trans. Signal Process., vol. 55, no. 5, pp. 2226-2238, May 2007, the MP equalizer described by Raviteja et al., “Interference Cancellation and Iterative Detection for Orthogonal Time Frequency Space Modulation,” in IEEE Transactions on Wireless Communications, vol. 17, no. 10, pp. 6501-6515, Oct. 2018, and the MP-1C-ML by Liu et al., “Method And Apparatus For Determining Symbols Transmitted Via Orthogonal Frequency Divisional Multiplex Signals”, International Patent Application No. PCT/EP2022/066642 (Publication No. WO/2022/268666)) and the 2S-BEM-DNN method are compared in Table III in FIG. 12. In Table III, N denotes OFDM frame length, n_iter denotes the number of iterations in the MP equalization, N_i[er denotes the number of iterations of channel estimation and equalization, M denotes modulation order, Q denotes number of BEM coefficients, (Q < 4), B_M denotes diagonal band width of the channel matrix H. The computational complexity of the DNN channel estimation is about 0(1001/ + 50L_o + 5000), where L, and L_o are the input and output sizes. As the complexity of the DNN channel estimation is much lower than equalization, only the complexity of equalizers is considered in Table III. For the MLSE equalizer, it was assumed that the channel matrix H is a banded diagonal matrix with diagonal band width of B_M. From Table III, it can be observed that the computational complexity of the 2S-BEM-DNN method grows only linearly with the OFDM frame length N. In contrast, the complexities of conventional MMSE and MP equalizers grow cubically and quadratically with N. For the MLSE equalizer, although its complexity grows only linearly with N, it grows exponentially with the diagonal band width B_M.

[0093] To visualize the complexity difference, the computational complexities of different equalization techniques are plotted in FIG. 13 for modulation order M = 2, 4 and 16. In the plots, it was assumed B_M = 20, Q = 4, N_iter = 4 and n_iter = 30.

Improvement on Performance

[0094] Simulation results for the 2S-BEM-DNN algorithm over DSC will now be discussed. The modulation scheme is Quadrature-phase shift keying (QPSK). The transmission bandwidth is 7.68MHz. The FFT size is N = 512. The DSC is of order L — 3, i.e., 4 multipaths are considered. Each channel tap is simulated as an i.i.d. random variable correlated in time according to Jakes’ model with the correlation function given as ]f2nnf_mafE_:), which is the zeroth-order Bessel function of the first land. The average total power of the 4 paths is normalized to 1. The extreme Doppler scenarios are considered in the simulation, where the vehicle speed is 500km/h and the carrier frequency f_c = 20GHz, which leads to normalized Doppler spread f_maxT_sN — 0.62. For the BEM model, K — 2N and Q — 4 are used. Using non-uniformly spaced frequencies, Q can be reduced from 4 to 2. For simplicity, various example embodiments assume uniformly spaced frequencies and Q = 4. The length of the pilot cluster is 2Q + 1 = 9 and the number of sub-blocks is L + 1 = 4. Therefore, the efficiency of the transmission is 93%. The parameters used in the simulation are summarized in Table IV shown in FIG. 14.

[0095] In the following, the performance of the 2S-BEM-DNN method is compared with a number of conventional methods, which include the iterative channel estimation and MLSE proposed by Barhumi et al., “MLSE and MAP equalization for transmission over doubly selective channels,” IEEE Trans. Veh. Technol., vol. 58, no. 8, pp. 4120-4128, Oct. 2009, the pilot-aided channel estimation and MMSE equalization proposed by Tang et al., “Pilot-assisted time-varying channel estimation for ofdm systems,” IEEE Trans. Signal Process., vol. 55, no. 5, pp. 2226-2238, May 2007, the distributed compressive sensing channel estimation and zero- forcing equalizer used by Qin et al., “Structured distributed compressive channel estimation over doubly selective channels,” m lEEE Transactions on Broadcasting, vol. 62, no. 3, pp. 521— 531, 2016, the MP equalizer described by Raviteja et al., “Interference Cancellation and Iterative Detection for Orthogonal Time Frequency Space Modulation,” in IEEE Transactions on Wireless Communications, vol. 17, no. 10, pp. 6501-6515, Oct. 2018, and the MP-IC-ML by Liu et al., “Method And Apparatus For Determining Symbols Transmitted Via Orthogonal Frequency Divisional Multiplex Signals”, PCT Patent Application No. PCT/EP2022/066642. [0096] The normalized mean square errors (NMSEs) of the channel estimation obtained by different channel estimation and equalization methods are shown in FIG. 15 for extreme Doppler scenarios. In particular, FIG 15 shows the comparison of the NMSE performance for different channel estimation and equalization methods, whereby f_c = 20GHz, vehicle speed = 500km/h, f_dT_sN — 0.62, efficiency = 93%. The NMSE is computed as:

(Equation 23) where h and h are the original and estimated time-domain channel coefficient vectors.

[0097] The BER obtained by different equalization schemes are shown in FIG. 16. The lower bound of BER is obtained by performing IC-MLSE only one time, and the IC and MLSE are based on the assumption that the channel state information (CSI) is perfectly known. For the banded-MMSE, B_M = 20 was used.

[0098] From FIGs 15 and 16, it can be observed that the 2S-BEM-DNN leads to significant performance improvement in terms of NMSE and BER, as compared with the conventional channel estimation and equalization methods at extreme Doppler scenario. The improvement is in terms of SNR over 10 dB. Furthermore, even without any iteration, the BER obtained by the channel estimation and equalization method is about the same as that obtained by the MP-1C- ML with iteration number = 4, and it is already very close to BER lower bounds.

[0099] FIG. 17 shows plots of comparison of performance of the 2S-BEM-DNN method trained and tested under the same or different SNRs, whereby f_c — 20GHz, vehicle speed = 500km/h, f_dT_sN = 0.62, efficiency = 93%. In FIG. 17, the BERs obtained by the 2S-BEM- DNN method trained and tested under same SNR or trained at a fixed SNR of 25 dB but tested under different SNRs are compared. It can be observed that when in the training of the DNN, the SNR is fixed to 25 dB, and in the test, the SNR varies from 5 to 25 dB, the BER performance of the 2S-BEM-DNN method only degrades slightly, as compared with DNN trained and tested under the same SNR. This result indicates that to simplify the use of DNN, it can be trained under high SNR or noiseless condition, and then used in noisy condition without making any change of it

[00100] FIG. 18 shows plots of comparison of performance of the 2S-BEM-DNN method trained and tested under the same or different Doppler (amount of pilot used = 7%, DNN pilot and data-aided channel estimation). In FIG. 18, the BERs obtained by the 2S-BEM-DNN method trained and tested under same Doppler or trained under extreme Doppler scenario (faT_sN — 0.62), tested under other Doppler scenarios (0.065 < f_dT_sN < 0.39) are compared. It can be observed that the DNN trained under extreme Doppler can still work well under other Doppler scenarios, and the BER achieved by the 2S-BEM-DNN method is very close to the lower bound in all the Doppler scenarios. This result indicates that to simplify the use of DNN, it can be trained under very high Doppler scenario, and then used in other lower Doppler scenarios without requiring any change.

[00101] Accordingly, various example embodiments provide a one-step two-stage channel estimation and equalization method (e.g., 2S-BEM-DNN) for OFDM transmission over fast changing frequency-selective channels. At the first stage of the detection, the method uses a first trained neural network model (e g., DNN) to perform a pilot-aided channel estimation and a banded-MMSE equalizer for symbol detection. At the second stage, the method uses a second trained neural network model (e g., DNN) together with the detected symbols obtained from the first stage to perform a data-aided channel estimation and an optimal maximum-likelihood (ML) equalizer, following an interference cancellation (IC) block, for symbol detection. Simulation results show that the 2S-BEM-DNN method achieves more than 10 dB improvement in terms of SNR, as compared with conventional channel estimation and equalization techniques, and without performing any iterations, and the BER achieved is very close to the BER obtained under perfect CSI. Furthermore, the complexity of the 2S-BEM- DNN method grows only linearly with the OFDM frame length N. Finally, the 2S-BEM-DNN method is also robust to the SNR and Doppler uncertainties. Accordingly, the 2S-BEM-DNN method according to various example embodiments may be implemented to achieve a one-step DNN receiver with linear complexity for extreme mobility OFDM communication. Accordingly, the 2S-BEM-DNN method according to various example embodiments may be employed in various practical applications, such as but not limited to, vehicular communications, high-speed train communications (e.g., having a speed of 600 km/h or faster), underwater acoustic communications, and low-earth-orbit (LEO) satellite communication, e g. StarLink, Furthermore, OFDM is the main waveform used in 4G/5G cellular standards and many WiFi standards. Therefore, the 2S-BEM-DNN method according to various example embodiments is also applicable to: 5G+ and 6G, due to backward compatibility requirements, mm-Wave cellular radio (e.g., Release 16, 30-60 GHz), new WiFi (e.g., 802. Had, 802.11bd, 60 GHz) and new joint radar communication system (e.g., 77 GHz, 120 GHz) standards.

[00102] While embodiments of the invention have been particularly shown and described with reference to specific embodiments, it should be understood by those skilled in the art that various changes in form and detail may be made therein without departing from the scope of the invention as defined by the appended claims. The scope of the invention is thus indicated by the appended claims and all changes which come within the meaning and range of equivalency of the claims are therefore intended to be embraced.

Claims

1. A method of receiving a transmitted signal over a time-varying frequency-selective channel, the method comprising: obtaining a received symbol signal in frequency domain based on the transmitted signal; performing a pilot-aided channel estimation with respect to the time-varying frequency- selective channel using a first trained neural network model based on the received symbol signal to obtain a plurality of first estimated channel coefficients; performing a first equalization based on the received symbol signal and the plurality of first estimated channel coefficients to obtain a plurality of first detected source symbols; performing a data-aided channel estimation with respect to the time-varying frequency- selective channel using a second trained neural network model based on the received symbol signal and the plurality of first detected source symbols to obtain a plurality of second estimated channel coefficients; and performing a second equalization based on the plurality of second estimated channel coefficients to obtain a plurality of second detected source symbols.

2. The method according to claim 1, wherein the time-varying frequency-selective channel is modeled based on a complex- exponential basis expansion model (CE-BEM), the plurality of first estimated channel coefficients is a plurality of first estimated BEM coefficients, and the plurality of second estimated channel coefficients is a plurality of second estimated BEM coefficients.

3. The method according to claim 1 or 2, wherein the pilot-aided channel estimation is performed using the first trained neural network model based on a plurality of pilot symbols and a channel estimation matrix determined based on the received symbol signal.

4. The method according to claim 3, wherein the first trained neural network model comprises: a first neural network portion configured to generate a plurality of initial first estimated channel coefficients in frequency domain based on the plurality of pilot symbols and the channel estimation matrix; a channel coefficient converting layer configured to convert the plurality of initial first estimated channel coefficients in frequency domain into a plurality of initial first estimated channel coefficients in time domain; and a second neural network portion configured to generate a plurality of refined first estimated channel coefficients in time domain, constituting the plurality of first estimated channel coefficients, based on the plurality of initial first estimated channel coefficients in time domain.

5. The method according to claim 4, wherein the plurality of pilot symbols is comprised in a pilot vector, and the first trained neural network model further comprises an input layer configured to convert the channel estimation matrix into a channel estimation one-dimensional (ID) array and concatenate the pilot vector and the channel estimation ID array to generate an input array to the first neural network portion.

6. The method according to claim 4 or 5, wherein the first neural network portion is trained based on a first objective function configured to minimize an error between the plurality of initial first estimated channel coefficients generated and a plurality of ground truth channel coefficients corresponding to the plurality of initial first estimated channel coefficients, and the second neural network portion is trained based on a second objective function configured to minimize an error between the plurality of refined first estimated channel coefficients generated and a plurality of ground truth channel coefficients corresponding to the plurality of refined first estimated channel coefficients.

7. The method according to any one of claims 1 to 6, wherein the second trained neural network model comprises: an input layer configured to generate a plurality of initial second estimated channel coefficients in frequency domain based on the received symbol signal and the plurality of first detected source symbols; a channel coefficient converting layer configured to convert the plurality of initial second estimated channel coefficients in frequency domain into a plurality of initial second estimated channel coefficients in time domain; and a neural network portion configured to generate a plurality of refined second estimated channel coefficients in time domain, constituting the plurality of second estimated channel coefficients, based on the plurality of initial second estimated channel coefficients in time domain.

8. The method according to claim 7, wherein the plurality of first detected source symbols comprises a plurality of detected data symbols and a plurality of detected pilot symbols, and the method further comprises replacing the plurality of detected pilot symbols in the plurality of first detected source symbols with a plurality of known pilot symbols corresponding to the plurality of detected pilot symbols.

9. The method according to claim 7 or 8, wherein the neural network portion of the second trained neural network model is trained based on a third objective function configured to minimize an error between the plurality of refined second estimated channel coefficients generated and a plurality of ground truth channel coefficients corresponding to the plurality of refined second estimated channel coefficients.

10. The method according to any one of claims 1 to 9, wherein the first equalization is based on a banded minimum mean square error equalization, and said performing the first equalization comprises: determining a first estimated channel matrix of the time-varying frequency- selective channel based on the plurality of first estimated channel coefficients; determining a banded first estimated channel matrix of the time-varying frequency-selective channel based on the first estimated channel matrix; and determining the plurality of first detected source symbols based on the received symbol signal and the banded first estimated channel matrix.

1 1 . The method according to any one of claims 1 to 10, further comprising: determining a second estimated channel matrix of the time-varying frequency-selective channel based on the plurality of second estimated channel coefficients; performing an inter-carrier interference removal with respect to the second estimated channel matrix to obtain an inter-carrier interference reduced second estimated channel matrix; and performing an inter-carrier interference removal with respect to the received symbol signal to obtain an inter-carrier interference reduced symbol signal.

12. The method according to claim 11, wherein the inter-carrier interference removal with respect to the received symbol signal is performed based on the received symbol signal, the plurality of first detected source symbols and the second estimated channel matrix.

13. The method according to claim 12, wherein the second equalization is based on a maximum-likelihood equalization and is performed based on the inter-carrier interference reduced symbol signal and the inter-carrier interference reduced second estimated channel matrix.

14. The method according to any one of claims 1 to 13, wherein said obtaining the received symbol signal comprises performing a discrete Fourier transform (DFT) based on the transmitted signal in time domain received to obtain the received symbol signal in frequency domain.

15. The method according to any one of claims 1 to 14, wherein the transmitted signal is transmitted over the time-varying frequency-selective channel based on orthogonal frequency division multiplexing (OFDM) transmission.

16. A receiver for receiving a transmitted signal over a time-varying frequency-selective channel, the receiver comprising: at least one memory; and at least one processor communicatively coupled to the at least one memory and configured to: obtain a received symbol signal in frequency domain based on the transmitted signal, perform a pilot-aided channel estimation with respect to the time-varying frequency- selective channel using a first trained neural network model based on the received symbol signal to obtain a plurality of first estimated channel coefficients; perform a first equalization based on the received symbol signal and the plurality of first estimated channel coefficients to obtain a plurality of first detected source symbols; perform a data-aided channel estimation with respect to the time-varying frequency- selective channel using a second trained neural network model based on the received symbol signal and the plurality of first detected source symbols to obtain a plurality of second estimated channel coefficients; and perform a second equalization based on the plurality of second estimated channel coefficients to obtain a plurality of second detected source symbols.

17. The receiver according to claim 16, wherein the time-varying frequency-selective channel is modeled based on a complex- exponential basis expansion model (CE-BEM), the plurality of first estimated channel coefficients is a plurality of first estimated BEM coefficients, and the plurality of second estimated channel coefficients is a plurality of second estimated BEM coefficients.

18. The receiver according to claim 16 or 17, wherein the pilot-aided channel estimation is performed using the first trained neural network model based on a plurality of pilot symbols and channel estimation matrix determined based on the received symbol signal.

19. The receiver according to claim 18, wherein the first trained neural network model comprises: a first neural network portion configured to generate a plurality of initial first estimated channel coefficients in frequency domain based on the plurality of pilot symbols and the channel estimation matrix; a channel coefficient converting layer configured to convert the plurality of initial first estimated channel coefficients in frequency domain into a plurality of initial first estimated channel coefficients in time domain; and a second neural network portion configured to generate a plurality of refined first estimated channel coefficients in time domain, constituting the plurality of first estimated channel coefficients, based on the plurality of initial first estimated channel coefficients in time domain.

20. The receiver according to claim 19, wherein the plurality of pilot symbols is comprised in a pilot vector, and the first trained neural network model further comprises an input layer configured to convert the channel estimation matrix into a channel estimation one-dimensional (ID) array and concatenate the pilot vector and the channel estimation ID array to generate an input array to the first neural network portion.

21. The receiver according to claim 19 or 20, wherein the first neural network portion is trained based on a first objective function configured to minimize an error between the plurality of initial first estimated channel coefficients generated and a plurality of ground truth channel coefficients corresponding to the plurality of initial first estimated channel coefficients, and the second neural network portion is trained based on a second objective function configured to minimize an error between the plurality of refined first estimated channel coefficients generated and a plurality of ground truth channel coefficients corresponding to the plurality of refined first estimated channel coefficients.

22. The receiver according to any one of claims 16 to 21, wherein the second trained neural network model comprises: an input layer configured to generate a plurality of initial second estimated channel coefficients in frequency domain based on the received symbol signal and the plurality of first detected source symbols; a channel coefficient converting layer configured to convert the plurality of initial second estimated channel coefficients in frequency domain into a plurality of initial second estimated channel coefficients in time domain; and a neural network portion configured to generate a plurality of refined second estimated channel coefficients in time domain, constituting the plurality of second estimated channel coefficients, based on the plurality of initial second estimated channel coefficients in time domain.

23. The receiver according to claim 22, wherein the plurality of first detected source symbols comprises a plurality of detected data symbols and a plurality of detected pilot symbols, and the at least one processor is further configured to replace the plurality of detected pilot symbols in the plurality of first detected source symbols with a plurality of known pilot symbols corresponding to the plurality of detected pilot symbols.

24. The receiver according to claim 22 or 23, wherein the neural network portion of the second trained neural network model is trained based on a third objective function configured to minimize an error between the plurality of refined second estimated channel coefficients generated and a plurality of ground truth channel coefficients corresponding to the plurality of refined second estimated channel coefficients.

25. The receiver according to any one of claims 16 to 24, wherein the first equalization is based on a banded minimum mean square error equalization, and said perform the first equalization comprises: determining a first estimated channel matrix of the time-varying frequency- selective channel based on the plurality of first estimated channel coefficients; determining a banded first estimated channel matrix of the time-varying frequency-selective channel based on the first estimated channel matrix; and determining the plurality of first detected source symbols based on the received symbol signal and the banded first estimated channel matrix.

26. The receiver according to any one of claims 16 to 25, wherein the at least one processor is further configured to: determine a second estimated channel matrix of the time-varying frequency-selective channel based on the plurality of second estimated channel coefficients; perform an inter-carrier interference removal with respect to the second estimated channel matrix to obtain an inter-carrier interference reduced second estimated channel matrix; and perform an inter-carrier interference removal with respect to the received symbol signal to obtain an inter-carrier interference reduced symbol signal.

27. The receiver according to claim 26, wherein the inter-carrier interference removal with respect to the received symbol signal is performed based on the received symbol signal, the plurality of first detected source symbols and the second estimated channel matrix.

28. The receiver according to claim 27, wherein the second equalization is based on a maximum-likelihood equalization and is performed based on the inter-carrier interference reduced symbol signal and the inter-carrier interference reduced second estimated channel matrix.

29. The receiver according to any one of claims 16 to 28, wherein said obtain the received symbol signal comprises performing a discrete Fourier transform (DFT) based on the transmitted signal in time domain received to obtain the received symbol signal in frequency domain.

30. The receiver according to any one of claims 16 to 29, wherein the transmitted signal is transmitted over the time-varying frequency-selective channel based on orthogonal frequency division multiplexing (OFDM) transmission.

31. A computer program product, embodied in one or more non-transitory computer- readable storage mediums, comprising instructions executable by at least one processor to perform a method of receiving a transmitted signal over a time-varying frequency-selective channel according to any one of claims 1 to 15.

32. A system for wireless communication over a time-varying frequency-selective channel, the system comprising: a transmitter configured to transmit a signal over the time-varying frequency-selective channel, and a receiver configured to receive the transmitted signal over the time-varying frequency- selective channel according to any one of claims 16 to 30.